How do I show the modifier stack panel when I work in the full screen area mode?
there is no defined "shortcut" to do that, and since blender window areas are by definition "non blocking" there is no way to let a "modifier stack panel" to jump in your fullscreen area mode, but you have alternatives, at least 2, imho:


reshape a layout to look as desired, ie calling it "fullscreen", set up like this:





use a separate window just for the "modifier stack panel", if you really wish it to go over your fullscreen area. You can duplicate the main window, and then reshape the secondary one to just show the "modifier stack panel", like this:




In linux it's easy to set this secondary floating "panel" as "always on top", but each OS has or can have similar utilities, so that you can always see it even when focus is on the background, fullscreen, area, like this: